(03006) Shafarevich's Conjecture for CY Manifolds I (Moduli of CY Manifolds)

来源:数学科学研究中心

Abstract:
In this paper we study the moduli spaces related to CY manifolds. We apply the results of this paper to the following problem. Let C be a fixed Riemann surface with fixed finite points on it. Fix the topological type of a CY manifold. Consider the set of all families of CY manifolds of the fixed topological type over C with degenerate fibres over the fixed points up to isomorphism. This set is called Shafarevich set. The analogue of Shafarevich conjecture for CY manifolds is for which topological types of CY the Shafarevich set is finite. It is well known that the analogue of Shafarevich conjecture is closely related to the study of the moduli space of polarized CY manifolds and the moduli space of the maps of fixed Riemann surface to the coarse moduli space of the CY manifolds. We prove the existence of the Teichm¨uller space of CY manifolds together with a universal family of marked CY manifolds. From this result we derive the existence of a finite cover of the coarse moduli space which is a non singular quasi-projective manifold. Over this cover we construct a family of polarized CY manifolds. We study the moduli space of maps of the fixed Riemann with fixed points on it to the moduli space of CY manifolds constructed in the paper such that the maps map the fixed points on the Riemann surface to the discriminant locus. If this moduli space of maps is finite then hafarevich conjecture holds. We relate the analogue of Shafarevich problem to the non vanishing of the Yukawa coupling. We give also a counter example of the Shafarevich problem for a class of CY manifolds.

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  • cmodliu.pdf